In this talk, I will discuss a connection between the ANEC (Averaged Null Energy Condition) operator and monotonicity of the renormalization group. In particular, I will show how the 2d c-theorem and 4d a-theorem can be derived using the ANEC. This derivation relies on contact terms appearing in specific ANEC correlators. I will also review a new infinite set of constraints that can be derived from the ANEC in 2d QFT. This program hints at a more general role for light-ray operators in QFT, which I will argue for.